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Theorem seqeq1 9922
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq1  |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )

Proof of Theorem seqeq1
StepHypRef Expression
1 iseqeq1 9919 . 2  |-  ( M  =  N  ->  seq M (  .+  ,  F ,  _V )  =  seq N (  .+  ,  F ,  _V )
)
2 df-seq3 9915 . 2  |-  seq M
(  .+  ,  F
)  =  seq M
(  .+  ,  F ,  _V )
3 df-seq3 9915 . 2  |-  seq N
(  .+  ,  F
)  =  seq N
(  .+  ,  F ,  _V )
41, 2, 33eqtr4g 2146 1  |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   _Vcvv 2620    seqcseq4 9912    seqcseq 9913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-iota 4993  df-fv 5036  df-oprab 5670  df-mpt2 5671  df-recs 6084  df-frec 6170  df-iseq 9914  df-seq3 9915
This theorem is referenced by:  seqeq1d  9925  seq3f1olemqsum  9990  iserex  10788  isumsplit  10946  ege2le3  11022
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